Can we make a robot ballerina perform a pirouette? Orbital stabilization of periodicmotions of underactuated mechanical systems

IFAC Proceedings Volumes (IFAC-PapersOnline)

Volumn 3, Issue PART 1, 2007, Pages 32-43

  Shiriaev, Anton S ^a,b   Freidovich, Leonid B ^a   Manchester, Ian R ^a  

^a UMEÅ UNIVERSITY   (Sweden)

^b NORWEGIAN UNIVERSITY OF SCIENCE AND TECHNOLOGY   (Norway)

Author keywords

Orbital stabilization; Periodic motions; Transverse linearization; Underactuated mechanical systems; Virtual holonomic constraints

Indexed keywords

DYNAMICAL SYSTEMS; LINEARIZATION; RICCATI EQUATIONS; ROBOTS; STABILIZATION; MECHANICAL ENGINEERING; SIGNAL PROCESSING;

ANALYSIS METHOD; GENERAL NONLINEAR SYSTEMS; ORBITAL STABILIZATION; PERIODIC MOTION; PERIODIC RICCATI EQUATION; STABILIZING CONTROL; UNDER-ACTUATED MECHANICAL SYSTEMS; VIRTUAL HOLONOMIC CONSTRAINTS; PERIODIC MOTIONS; TRANSVERSE LINEARIZATION; UNDERACTUATED MECHANICAL SYSTEMS;

MECHANICS;

EID: 79961005615     PISSN: 14746670     EISSN: None     Source Type: Conference Proceeding    
DOI: 10.3182/20070829-3-ru-4911.00005     Document Type: Conference Paper

References (57)

1. Aguilar, L., I. Boiko, L. Fridman and R. Iriarte (2006). Generation of periodic motions for underactuated mechanical system via secondorder sliding-modes. In: Proc. of American Control Conference. Minneapolis, MA, USA. pp. 56-98.
2. Altafini, C. (2002). Following a path of varying curvature as an output regulation problem. IEEE Trans. Autom. Control 47(9), 1551-1556.
3. Andronov, A.A. and A. Vitt (1933). On Lyapunov stability. (In Russian) Experimental Theoretical Physics 3, 373-374.
4. Aracil, J., F. Gordillo and J.A. Acosta (2002). Stabilization of oscillations in the inverted pendulum. In: Proc. of the 15th IFAC World Congress. Barcelona, Spain.
5. Banaszuk, A. and J. Hauser (1995). Feedback linearization of transverse dynamics for periodic orbits. Systems and Control Letters 26, 95-105.
6. Bittanti, S., P. Colaneri and G. de Nicolau (1991). The periodic Riccati equations. In: The Riccati Equation (S. Bittanti, A. Laub and J.C. Willems, Eds.). pp. 127-162. Springer. Berlin, Germany.
7. Bloch, A.M., N.E. Leonard and J.E.Marsden (2000). Controlled Lagrangians and the stabilization of mechanical systems. i. the first matching theorem. IEEE Transactions on Automatic Control 45(12), 2253-2270. (Pubitemid 33775429)
8. Byrnes, C.I., A. Isidori and J.C. Willems (1991). Passivity, feedback equivalence, and the global stabilization of minimum phase nonlinear systems. IEEE Transactions on Automatic Control 36, 1228-1240.
9. Canudas-de-Wit, C., B. Espiau and C. Urrea (2002). Orbital stabilization of underactuated mechanical systems. In: Proc. of the 15th IFAC World Congress. Barcelona, Spain.
10. Chevallereau, C., A. Formal'sky and D. Djoudi (2004). Tracking a joint path for the walk of an underactuated biped. Robotica 22(1), 15-28.
11. Chevallereau, C., E.R. Westervelt and J.W. Grizzle (2005). Asymptotically stable running for a five-link, four-actuator, planar, bipedal robot. International Journal of Robotics Research 24(6), 431-464. (Pubitemid 40718615)
12. Chevallereau, C., G. Abba, Y. Aoustin, F. Plestan, E.R. Westervelt, C. Canudas-de-Wit and J.W. Grizzle (2003). Rabbit: A testbed for advanced control theory. IEEE Control Systems Magazine 23(5), 57-79.
13. Chung, C.C. and J. Hauser (1997). Nonlinear h1 control around periodic orbits. Systems and Control Letters 30(2), 127-137. (Pubitemid 127451520)
14. Coelho, P. and U. Nunes (2003). Lie algebra application to mobile robot control: a tutorial. Robotica 21, 483-493.
15. Fossen, T.I. (2002). Marine Control Systems. Marine Cybernetics.
16. Fossen, T.I. and J.P. Strand (2001). Nonlinear passive weather optimal positioning control for ships and rigs: experimental results. Automatica 37, 701-715. (Pubitemid 32264798)
17. Fradkov, A.L. and A.Yu. Pogromsky (1998). Introduction to control of oscillations and chaos. World Scientific. Singapore.
18. Freidovich, L., A. Robertsson, A. Shiriaev and R. Johansson (2007a). Stable periodic motions of inertia wheel pendulum via virtual holonomic constraints. In: Proc. of European Control Conference. Kos, Greece.
19. Freidovich, L., A. Robertsson, A. Shiriaev and R. Johansson (2008). Periodic motions of the Pendubot via virtual holonomic constraints: theory and experiment. Automatica.
20. Freidovich, L., A. Shiriaev and I. Manchester (2007b). Experimental implementation of stable oscillations of the Furuta pendulum around the upward equilibrium. In: Proc. Of IEEE/RSJ International Conference on Intelligent Robots and Systems. San Diego, CA, USA.
21. Freidovich, L., A. Shiriaev and I. Manchester (2007c). Transitions between limit cycles for an underactuated system: virtual constraints approach. In: Proc. of the 7th IFAC Sympo-sium on Nonlinear Control Systems. Pretoria, South Africa.
22. Freidovich, L., A. Shiriaev, F. Gomez-Estern, F. Gordillo and J. Aracil (2007d). Partialenergy- shaping control for orbital stabilization of high frequency oscillations of the Furuta pendulum. In: Proc. of the 46th Confer- ence on Decision and Control. New Orleans, USA.
23. Gillespie, R.B., J.E. Colgate and M.A. Peshkin (2001). A general framework for cobot control. IEEE Transactions on Robotics and Automation 17(4), 391-401. (Pubitemid 33048325)
24. Grizzle, J.W., C.H. Moog and C. Chevallereau (2005). Nonlinear control of mechanical systems with an unactuated cyclic variable. IEEE Transactions on Automatic Control 50(5), 559-576. (Pubitemid 40784429)
25. Grizzle, J.W., G. Abba and F. Plestan (1999). Proving asymptotic stability of a walking cycle for a five dof biped robot model. In: Proc. of the 2nd International Conference on Climbing and Walking Robots. Portsmouth, UK.
26. Grizzle, J.W., G. Abba and F. Plestan (2001). Asymptotically stable walking for biped robots: Analysis via systems with impulse effects. IEEE Transactions on Automatic Control 46(1), 51-64. (Pubitemid 32191020)
27. Gusev, S., A. Shiriaev and L. Freidovich (2007). LMI approach for solving periodic matrix Riccati equation. In: Proc. of the 3rd IFAC Workshop Periodic Control Systems. St. Petersburg, Russia.
28. Hale, J.K. (1980). Ordinary Differential Equa-tions. Krieger, Malabar. FL, USA.
29. Hauser, J. and C.C. Chung (1994). Converse Lyapunov function for exponential stable periodic orbits. Systems and Control Letters 23, 27-34.
30. Johansson, S., B. Kågström, A. Shiriaev and A. Varga (2007). Comparing one-shot and multi-shot methods for solving periodic Riccati differential equations. In: Proc. of the 3rd IFAC Workshop Periodic Control Systems. St. Petersburg, Russia.
31. Khalil, H.K. (2002). Nonlinear systems. 3 ed. Prentice Hall.
32. Leonov, G.A. (2006). Generalization of the Andronov-Vitt theorem. Regular and chaotic dynamics 11(2), 281-289.
33. Manchester, I.R., A.S. Shiriaev and A.V. Savkin (2007). On motion planning for an underactuated ship: Fundamental limitations and a bearings-only navigation strategy. In: Proc. Of the 46th Conference on Decision and Control. New Orleans, USA.
34. Mazenc, F. and S. Bowong (2003). Tracking trajectories of the cart-pendulum system. Automatica 39(4), 677-684.
35. Miossec, S. and Y. Aoustin (2005). A simplified stability study for a biped walk with underactuated and overactuated phases. International Journal of Robotics Research 24(7), 537-551. (Pubitemid 40963749)
36. Nakaura, S., Y. Kawaida, T. Matsumoto and M. Sampei (2004). Enduring rotatory motion control of devil stick. In: Proc. of the 6th IFAC Symposium on Nonlinear Control Systems. Stuttgart, Germany.
37. Nam, K. and A. Arapostathis (1992). A sufficient condition for local controlability of nonlinear systems along closed orbits. IEEE Transactions on Automatic Control 37(3), 378-380.
38. Nielsen, C. and M. Maggiore (2006). Further results on transverse feedback linearization of multi-input systems. In: Proc. of the 45th Conference on Decision and Control. San Diego, USA.
39. Ortega, R., A. Loria, P.J. Nicklasson and H. Sira-Ramirez (1998). Passivity-based control of Euler-Lagrange systems. Communications and Control Engineering. Springer- Verlag. Berlin, Germany.
40. Ortega, R., M. Spong, F. Gomez and G. Blankenstein (2002). Stabilization of underactuated mechanical systems via interconnection and damping assignment. IEEE Transactions on Automatic Control.
41. Perram, J., C. Canudas-de-Wit A. Shiriaev and F. Grognard (2003). Explicit formula for a general integral of motion for a class of mechanical systems subject to holonomic constraint. In: Proc. of the 2nd IFAC Workshop Control Methods for Lagrangian and Hamil- tonian Systems. Seville, Spain.
42. Poincaré, H. (1916-1954). Oeuvres comleés. Vol. 11. Gauthier-Villars. Paris, France.
43. Rouche, N. and J. Mawhin (1980). Ordinary differential equations. Stability and periodic solutions. Pitman Publishing ltd.
44. Samson, C. (1995). Control of chained systems: application to path following and timevarying point-stabilization of mobile robots. IEEE Transactions on Automatic Control 40(1), 64-77.
45. Shimizu, T., S. Nakaura and M. Sampei (2006). The control of bipedal running robot based on output zeroing considered the rotation of the ankle joint. In: Proc. of the 45th Conference on Decision and Control. San Diego, USA.
46. Shiriaev, A., L. Freidovich, A. Robertsson and R. Johansson (2006a). Generating stable propeller motions for devil stick. In: Proc. Of the 3rd IFAC Workshop Lagrangian and Hamiltonian Methods for Nonlinear Control. Nagoya, Japan.
47. Shiriaev, A., L. Freidovich, A. Robertsson, R. Johansson and A. Sandberg (2007). Virtualconstraints- based design of stable oscillations of Furuta pendulum: theory and experiments. IEEE Transactions on Robotics 23(4), 827-832. (Pubitemid 47308121)
48. Shiriaev, A.S., A. Robertsson, J.W. Perram and A. Sandberg (2006b). Periodic motionplanning for virtually constrained mechanical system. Systems and Control Letters55(11), 900-907. (Pubitemid 44344831)
49. Shiriaev, A.S., J.W. Perram and C. Canudasde- Wit (2005). Constructive tool for orbitalstabilization of underactuated nonlinear systems: virtual constraints approach.IEEE Transactions on Automatic Control 50(8), 1164-1176. (Pubitemid 41410114)
50. Shiriaev, A.S., L.B. Freidovich and I.R. Manchester (n.d.). Transverse linearization for mechanical systems with underactuation degree one. Submitted.
51. Skjetne, R., T.I. Fossen and P.V. Kokotović (2004). Robust output maneuvering for a class of nonlinear systems. Automatica 40, 373-383.
52. Spong, M.W. (1997). Underactuated mechanical systems. In: Control Problems in Robotics and Automation (B. Siciliano and K.P. Valavanis, Eds.). Vol. 230 of Lecture Notes in Control and Inf. Sci. Springer-Verlag. London, UK.
53. Spong, M.W. (2004). Partial feedback linearization of underactuated mechanical systems. In: Proc. of International Conference on Intelligent Robots and Systems. Munich, Germany.
54. Spong, M.W., S. Hutchinson and M. Vidyasagar (2006). Robot modeling and control. John Wiley and Sons.
55. Urabe, M. (1967). Nonlinear autonomous oscillations. Academic Press. N.Y., USA.
56. Yakubovich, V.A. (1986). Linear quadratic optimization problem and frequency theorem for periodic systems. (In Russian) Siberian Mathematical Journal 27(4), 181-200.
57. Yoshizawa, T. (1966). Stability theory by Liapunov's second method. he mathematical society of Japan. Tokyo, Japan.